What question did this study set out to answer?

To articulate the logical framework of the SIARC series focused on polynomial continued fractions and their classifications.

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May 1, 2026Open Access

An Arithmetic Stratification of Polynomial Continued Fractions: Program Statement of the SIARC Series

Key Points

To articulate the logical framework of the SIARC series focused on polynomial continued fractions and their classifications.
Defined a four-tier hierarchy including Algebraic, Transcendental, Trans-stratum, and Barrier.
Presented the Trans-Stratum−2/9 Conjecture in a precise format.
Listed open problems and the roles of companion papers in the project.
Established a citable structure for the SIARC series without proving any claims.
Identified each claim as a conjecture or a reference to a related companion paper.
Provided a unified notation system for better coherence across the series.

Abstract

Thispaper isaprogramstatement,notaresultspaper. Itarticulatesthelogical structureoftheSIARCseries: abodyof6–8companionpapersthattogetherproposeanarithmeticstratificationofpolynomial continuedfractions(PCFs)organizedbytheunderlyinguniformizingFuchsiangroupandbyanobstructiontier.Westatethe four-tierhierarchy(Algebraic/Transcendental/Trans-stratum/Barrier),presenttheTrans-Stratum−2/9Conjecture inprecise form, listtheopenproblemsthatanchortheprogram,andtabulatetheroleandcurrentstatusofeachcompanionpaper.Noproofsarereproducedhere;everynontrivial claimis either statedasaconjectureor remarkeduponwithapointer totheappropriatecompanionpaper, citedbyZenodoDOIwhereavailable. Thepurposeof thisdocument is toprovideastable, citable scope-anchor for theprogramandtofixunifiednotationacrosstheseries.

An Arithmetic Stratification of Polynomial Continued Fractions: Program Statement of the SIARC Series

Key Points

Abstract

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